Activity 2.1.5 Circuit Simplification: DeMorgan’s Theorems Introduction Despite all of the work done by George Boole, there was still more work to be done. Expanding on Boole’s studies, Augustus DeMorgan (1806-1871) developed two additional theorems that now bear his name. Without DeMorgan’s Theorems, the complete simplification of logic expression would not be possible. Theorem #1: XY X Y Theorem #2 : X Y XY As we will seen in later activities, DeMorgan’s Theorems are the foundation for the NAND and NOR logic gates. In this activity you will learn how to simplify logic expressions and digital logic circuits using DeMorgan’s two theorems along with the other laws of Boolean algebra. Procedure Using DeMorgan’s theorems and the other theorems and laws of Boolean algebra, simplify the following logic expressions. Note the theorem/law used at each simplification step. Be sure to put your answer in Sum-Of-Products (SOP) form. 1. F1 X Y F1= 𝑋𝑌̅ Theorems 21 and 9. I then rewrote the equation w/o AND symbols. 2. F2 ( R S ) ( T U ) ̅ F2= 𝑅̅ 𝑆 + 𝑇𝑈 Theroem #20 and #9. 3. F3 M N ̅ F3= 𝑀 + 𝑁 Theorem #9. © 2014 Project Lead The Way, Inc. Digital Electronics Activity 2.1.5 Circuit Simplification: DeMorgan’s Theorems – Page 1 4. F4 W X ( Y Z ) F4= (𝑊 + 𝑋̅) + 𝑌𝑍̅ Theorem #20 and #9. 5. F5 P Q R P Q R F5= (𝑃𝑄̅ 𝑅) (𝑃̅𝑄𝑅̅ ) Theorem #21 and #9 6. F6 W X Y Z F6= 𝑊𝑋̅𝑌𝑍 Let’s see how we would utilize DeMorgan’s theorems to simplify a digital logic circuit. 7. Write the UN-SIMPLIFIED logic expression for the output Do-Nothing in the logic circuit shown below. ̅̅̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅ ̅ + 𝑌) + (𝑌𝑍)) + (𝑋 ((𝑋 + 𝑌̅) The output is call Do-Nothing because that is exactly what the circuit does, nothing; yet it’s a good example for learning about DeMorgan’s theorems. © 2014 Project Lead The Way, Inc. Digital Electronics Activity 2.1.5 Circuit Simplification: DeMorgan’s Theorems – Page 2 8. Using DeMorgan’s theorems and the other theorems and laws of Boolean algebra, simplify the logic expression Do Nothing. Be sure to put your answer in Sum-OfProducts (SOP) form. ̅̅̅̅) + (𝑌𝑍) (𝑋𝑌 9. In the space provided, draw an AOI circuit that implements the simplified logic expression Do Nothing. For this implementation you may assume that AND & OR gates are available with any number of inputs. Do Nothing – I 10. Re-implement the circuit assuming that only 2-input AND gates (74LS08), 2-input OR gates (74LS32), and inverters (74LS04) are available. Draw this circuit in the space provided. © 2014 Project Lead The Way, Inc. Digital Electronics Activity 2.1.5 Circuit Simplification: DeMorgan’s Theorems – Page 3 Do Nothing – II Conclusion 1. Draw the gate equivalent for DeMorgan’s two theorems. Theorem #1: XY X Y © 2014 Project Lead The Way, Inc. Digital Electronics Activity 2.1.5 Circuit Simplification: DeMorgan’s Theorems – Page 4 Theorem #2 : X Y XY 2. How would you prove that the original Do Nothing circuit and the simplified version are equivalent? You could create a truth table for them & you could build the actual circuit in order to prove their similatiry. 3. If each GATE cost 5¢ and you made 100,000 of the unsimplified units, how much of the company’s money did you waste on the unsimplified Do-Nothing project? They lost due to the unsimplified Do-Nothing $30,000. © 2014 Project Lead The Way, Inc. Digital Electronics Activity 2.1.5 Circuit Simplification: DeMorgan’s Theorems – Page 5